Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots.
Answer: If $2x^2 + 5x + b = 0$ has two rational solutions, then its discriminant, $5^2 - 4 \cdot 2 \cdot b = 25 - 8b$, must be a perfect square. Since $b$ is positive, it follows that $25 - 8b \ge 0 \Longrightarrow b \in \{1,2,3\}$. Checking each one, we see that $b = 2$ and $b = 3$ indeed work, and their sum is $2 + 3 = \boxed{5}$.